On the variation of maximal operators of convolution type II

TL;DR

This paper proves that certain maximal convolution operators related to elliptic and parabolic equations diminish variation, with applications on Euclidean space, torus, and sphere, based on their subharmonicity in specific regions.

Contribution

It establishes the variation-diminishing property of maximal operators of convolution type associated with elliptic and parabolic equations across different geometric settings.

Findings

Maximal operators are variation-diminishing in Euclidean space, torus, and sphere.

These operators are subharmonic in their detachment sets.

The study extends the understanding of regularity properties of convolution-based maximal functions.

Abstract

In this paper we establish that several maximal operators of convolution type, associated to elliptic and parabolic equations, are variation-diminishing. Our study considers maximal operators on the Euclidean space $\mathbb{R}^d$, on the torus $\mathbb{T}^d$ and on the sphere $\mathbb{S}^d$. The crucial regularity property that these maximal functions share is that they are subharmonic in the corresponding detachment sets.